The PE of a certain spring when streched from natural length through a distance `0.3m` is `5.6J`. Find the amount of work in joule that must be done on this spring to stretch it through an additional distance `0.15m`.

Explanation

To solve the problem, we will use the concept of potential energy stored in a spring, which is given by the formula:

\[ PE = \frac{1}{2} k x^2 \]

where \( PE \) is the potential energy, \( k \) is the spring constant, and \( x \) is the extension from the natural length of the spring.

### Step 1: Find the spring constant \( k \)

We know that the potential energy when the spring is stretched by \( 0.3 \, m \) is \( 5.6 \, J \). We can use this information to find \( k \).

\[ 5.6 = \frac{1}{2} k (0.3)^2 \]

Calculating \( (0.3)^2 \):

\[ (0.3)^2 = 0.09 \]

Now substituting back into the equation:

\[ 5.6 = \frac{1}{2} k (0.09) \]

Multiplying both sides by \( 2 \):

\[ 11.2 = k (0.09) \]

Now, solving for \( k \):

\[ k = \frac{11.2}{0.09} \approx 124.44 \, N/m \]

### Step 2: Calculate the work done to stretch the spring an additional \( 0.15 \, m \)

The total extension when the spring is stretched by an additional \( 0.15 \, m \) will be:

\[ x_{final} = 0.3 + 0.15 = 0.45 \, m \]

Now we can find the potential energy at this new extension:

\[ PE_{final} = \frac{1}{2} k (0.45)^2 \]

Calculating \( (0.45)^2 \):

\[ (0.45)^2 = 0.2025 \]

Now substituting \( k \) and \( (0.45)^2 \) into the equation:

\[ PE_{final} = \frac{1}{2} (124.44)(0.2025) \approx 12.6 \, J \]

### Step 3: Calculate the change in potential energy

Now we find the change in potential energy, which is equal to the work done:

\[ \Delta PE = PE_{final} - PE_{initial} \]

Where \( PE_{initial} \) is \( 5.6 \, J \):

\[ \Delta PE = 12.6 - 5.6 = 7 \, J \]

### Final Answer

The amount of work that must be done on the spring to stretch it through an additional distance of \( 0.15 \, m \) is \( 7 \, J \).

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